Texture mirror in paint mode
You can paint in Texture Paint mode and toggle symmetry in the Symmetry panel for X, Y and Z axis.

If you want to just mirror the texture, you can delete one half of the model and use a mirror modifier, unwrap the mesh to align with your intended texture and then apply the mirror modifier to get the mesh UVs to align on both sides.

edit: To use the mirror modifier to align both sides to a single side of paint image, I first delete and add a mirror modifier. I unwrap and arrange my UVs to get the most pixels in the image bounds. I set a new image with a grid to check the density and see if it is even, also still looking for stretching. I apply the mirror and set up a new image at the resolution I want to paint from the Slots tab in the texture toolbar. The painting then is all mirrored across the x axis here.